Nuprl Lemma : rmul-rdiv-cancel5
∀[a,b,c:ℝ]. ((b/a) * (a/c)) = (b/c) supposing a ≠ r0 ∧ c ≠ r0
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
false: False,
implies: P ⇒ Q,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
rtermMultiply: left "*" right,
pi2: snd(t),
prop: ℙ
Lemmas referenced :
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermVar_wf,
int-to-real_wf,
istype-int,
req_witness,
rmul_wf,
rdiv_wf,
rneq_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
lambdaEquality_alt,
int_eqEquality,
hypothesisEquality,
independent_isectElimination,
approximateComputation,
sqequalRule,
independent_pairFormation,
independent_functionElimination,
productIsType,
universeIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[a,b,c:\mBbbR{}]. ((b/a) * (a/c)) = (b/c) supposing a \mneq{} r0 \mwedge{} c \mneq{} r0
Date html generated:
2019_10_29-AM-09_55_12
Last ObjectModification:
2019_04_01-PM-07_05_36
Theory : reals
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